Transcendental inverse eigenvalue problems in damage parameter estimation

More efficient use of determinants to solve transcendental structural eigenvalue problems reliably

Detection of multiple cracks using frequency measurements

Purpose Modelling methods can be helpful for understanding vibrations of beam structures including cracks, as well as for early detection of crack. This study aims to provide an analytical modelling approach for a cantilever beam considering a slant vertical crack along its height. However, previous uniform crack methods cannot be used for describing this case. The results from the analytical, finite element (FE) and experimental methods are compared to verify the vibration problem. Design/methodology/approach A massless rotational spring model is adopted to describe the crack. An extended method based on the calculation method for a uniform vertical edge crack is proposed to obtain the stiffness of the slant case. The beam is divided into a series of independent thin slices along the beam height. An Euler–Bernoulli beam model is applied to formulate each slice. The crack in each slice is considered as a uniform one. The transfer matrix method in the literature is used to obtain the beam vibration frequencies and mode shapes. Influences of crack location and sizes on the natural frequencies for the cantilever beam, as well as the mode shapes, are analysed. An established FE model and test results in the listed references are used to validate the developed method. Findings The numerical results show that the rotational stiffness at the cracked section and the natural frequencies of the beam decrease by increasing the crack sizes; the natural frequencies for the beam are greatly influenced by the crack sizes and location; the first natural frequency decreases with the distance from the beam fixed end to the crack location; the value of the first natural frequency reaches a minimum value when the crack is at the beam fixed end; the value of the second natural frequency is a minimum value when the crack is at the beam middle; and the value of the third natural frequency is a minimum value when the crack is at the beam free end. Saltation is observed in some mode shapes at the crack location, especially for larger crack depths; but, the mode shapes of the beam are slightly influenced by the vertical crack. Originality/value This study gives a useful analytical modelling method for free vibration analysis for the cantilever beam with a vertical crack, which can overcome the disadvantages of the previous uniform crack methods.

The solution of an infinite elastic plane containing a macro-crack and a cluster of micro-cracks is presented based on Muskhelishvili’s complex potential method. A step-by-step sub-problem procedure is used to satisfy the stress boundary conditions on each crack surface. A local damage parameter expressed by stress intensity factor is presented to evaluate the damage magnitude. Three damage configurations as parallel, radiating and arbitrarily oriented micro-cracks have been designed to simulate the damage around the macro-crack tip. The results show the micro-cracks located in front of the macro-crack tip have the amplifying effect on macro-crack growth, and the amplifying effect increases with the number of micro-cracks increase. Moreover, when the macro-crack length decreases or the distance between micro-cracks and macro-crack increases, the damage parameter of macro-crack and micro-cracks will decrease. It is found that the largest damage is the damage configuration of arbitrarily oriented micro-cracks among three damage configurations.

This paper investigates the free vibration of a homogeneous Euler–Bernoulli beam with multiple transverse cracks under non-symmetric boundary conditions. The differential equation is formulated by introducing Dirac’s delta function into the uniform flexural stiffness, and the close-form solution of mode shapes is then derived by applying the Laplace transform technique. The proposed method is validated against existing experimental method for damaged cantilever beams. With the validated method, a parametric study is performed to study the effect of crack numbers, damage parameters and crack locations on the natural frequencies and mode shapes for three non-symmetric boundary conditions (pinned–clamped, clamped–free shear and pinned–free shear).

The inverse matrix eigenvalue problem is the problem of reconstruction of a matrix (determination of the unknown parameters of a certain matrix) for a given finite set of spectral data (points of the spectrum). The spectral data used in this case may contain complete or only partial information on the eigenvalues or eigenvectors. The aim of an inverse eigenvalue problem is to construct a matrix that preserves a certain specific structure and the prescribed spectral properties. There are two basic questions connected with any inverse eigenvalue problem: the theoretical question on the possibility of construction of its solutions and the practical question of its determination. The main investigations connected with solvability deal with establishing necessary or sufficient conditions under which the inverse eigenvalue problem is solvable. On the other hand, the main difficulty in calculations is connected with the development of a numerical procedure which would enable us to construct the matrix itself according to the a-priori specified spectral data. Both questions are important and complicated. Numerous works are devoted to the conditions of existence and uniqueness of solutions for different formulations of inverse matrix spectral problems and the methods used for their solution (see, e.g., [1–3, 7, 9–13, 15] and the literature therein). In the present work, we consider a numerical algorithm used for the solution of inverse eigenvalue problems under the assumption that this solution exists. Hence, we consider inverse algebraic eigenvalue problems.

A crack identification algorithm based on a mathematical model has been developed to identify crack location and depth in stepped cantilever Euler—Bernoulli beam carrying concentrated masses. In order to estimate crack location and depth in the beam the proposed algorithm utilizes the variation of the difference between the natural frequencies of cracked and intact systems versus single mass location along the beam span. The assumed mode method is used to derive the mathematical model for the system under investigation, in which the crack's effect is introduced to the system as a global effect. The advantage of the proposed algorithm is to identify the crack by monitoring a single natural frequency of the system. The algorithm can utilize the measurements of the first few system natural frequencies to check/reconfirm its identification results. Furthermore, it is efficient and can be simply implemented for any structure that has a slowly moving mass, as in the case of overhead gantry and girder cranes. The identification algorithm is tested using the first system's natural frequency obtained from an experimental work as well as from a finite element analysis. Moreover, the algorithm is tested once more using the second system's natural frequency obtained from finite element analysis. The tests' results show that the crack depth and location can be predicted with sound accuracy.

Summary Inverse eigenvalue problems associated with self-adjoint differential equations of order two or higher are considered. The question of what kind of data are necessary and sufficient to insure the existence of a unique solution is examined. A method of solution for well-posed inverse eigenvalue problems is then presented. The implications of this analysis to the inverse problem for the Earth's spheroidal modes of vibration is also discussed. In this talk, I propose to present a partial survey of the mathematical literature on inverse eigenvalue problems, with an eye toward geophysical applications, and to state some new results which I obtained recently. Perhaps the inverse eigenvalue problem which is best known to geophysicists is that related to the normal modes of vibration of the Earth. It can be stated thus: assuming that the Earth is spherically symmetric and given the natural frequencies ,S, and ,T, of the spheroidal and torsional modes of vibration, can we deduce the density p(r) and the speed of propagation Vr Takeuchi & Saito 1972) and very little is known about inverse eigenvalue problems of order higher than two. In order to gain an understanding of inverse eigenvalue problems one should perhaps divorce oneself from a specific problem and investigate rather a sequence of problems of increasing complexity. If you allow me to adopt this attitude, then the first task would be a rough classification of eigenvalue problems. Without pretending to be exhaustive, let me divide eigenvalue problems into two classes depending upon whether they are associated with ordinary or partial differential equations, i.e. depending upon whether they are related to one-dimensional or multi-dimensional problems. The following Sturm-Liouville problem: u+{n-q(x)}u = 0

Most of the rotating machines used in process industries or in manufacturing plant need maintenance and repair. However, failure of just one of these machines can disturb an entire process with losses in terms of production, manpower, and equipment repair or replacement. Also failure of a single machine component in process industries like petrochemicals or power stations can result in the loss of millions of rupees per down time hour. These facts together with higher costs for new equipment have placed increased demand on plant maintenance to keep existing equipment operating efficiently with higher availability. A number of non-destructive crack detection techniques have been developed, such as ultrasonic testing, X-ray technique, magnetic particle method etc. Every method has some advantages and disadvantages. In recent years, ultrasonic testing has gained greater attention for monitoring the cracks in structures and machine components. Most of these methods are very laborious and time consuming, in the case of larger components like bridges, long pipe lines, railway tracks etc. These inconveniences can be avoided by the use of vibration monitoring technique such as model analysis. Therefore the development of vibration monitoring techniques has received increasing attention in recent years. Damage in structure alters its dynamic characteristics. The change is characterized by a change in model parameters i.e. model frequencies, model damping values and mode shapes associated with each model frequencies. Changes also occur in some of the structural parameters like mass, damping, stiffness and flexibility matrices of structure. Thus a vibration technique can be suitably used as a nondestructive test for crack detection of components to be tested. In this paper, theoretical and experimental analysis of a vibrating cantilever beam with a crack is carried out. This method is used to address the inverse problem of assessing the crack location and crack size in various beam structure. The method is based on measurement of natural frequencies, which are global parameter and can be easily measured from any point on the structure. In theoretical analysis the crack is simulated by an equivalent spring, connecting the two segments of the beam. Analysis of this approximate model results in algebraic equation, which relates the natural frequencies of the beam, and crack location. Also the relationship between the natural frequencies, crack location and crack size has also been developed. For identification of crack location and crack size, it was shown that data on the variation of the first two natural frequencies are sufficient. The computation of natural frequencies of uncracked and cracked beam is facilitated by a finite element method package. This database is to be utilized in an analytical method to address the inverse problem to identify the crack location and crack size.

In this work, we present a two-step procedure for damage identification in beam structures exploiting modal curvature changes. The reconstruction of modal curvatures requires the knowledge of several mode shape components along the analyzed beam. This requirement is practically unachievable when mode shapes are identified via vibration-based monitoring using a limited number of accelerometers. To overcome this limitation, in the first step of the proposed procedure, we perform a mode shape expansion employing a reduced subset of measured modal components. The remaining measured components are used as control parameters to formulate a first hypothesis on damage location and extent. For this purpose, the expansion procedure is performed considering a number of possible damage scenarios, consisting of a location and a severity (loss of stiffness) of the damage. Using the Total modal assurance criterion (TMAC), we select the expanded modes with the highest degree of correspondence with the measured control components. These expanded modes are thus associated with a first guess of the damage location and severity. In the next step, this initial damage identification is verified through the computation of a modal curvature-based damage index. If the curvature-based damage identification confirms the previous identification, the damage location and extent are determined. The procedure can be easily extended to identify multiple simultaneously damaged elements. The approach is numerically validated using a benchmark beam modeled via finite elements, investigating the influence of different parameters such as noise, position of the control components and beam discretization on the identification success rate. Finally, the procedure is tested on two experimental specimens: a steel beam, with three different damage configurations and a concrete beam progressively damaged with multiple damage locations.

The fatigue life evaluation of adhesively bonded joints has attracted more and more attention with its extensive application. Here, the fatigue experiments under various loading conditions were carried out to understand the stress amplitude, mean stress, cycle period, and loading path related to the fatigue life of the adhesively bonded tubular butt-joints. The experimental results showed that the fatigue life of the adhesively bonded butt-joint decreased with the increase of stress amplitude and mean stress, and the influence of cycle period on the fatigue life was more obvious since the mechanical property of the selected adhesive was rate dependent. Meanwhile, the fatigue life of the adhesively bonded butt-joint was influenced by the loading path under multiaxial loading conditions. In addition, the relationship between the damage parameters and fatigue life was characterized based on the experimental observations. Accordingly, a stress-based fatigue life prediction model was developed which can simultaneously consider multiple damage parameters related to fatigue life of the adhesively bonded joint. The predicted results indicated that the predicted life correlated well with the experimental data of the adhesively bonded joints.

A new iteratives procedure is proposed to solve the transcendental eigenvalue problem of the “exact” dynamic stiffness matrix of linearly elastic plane frames. According to the presented algorithm the exact solution, i.e. natural frequencies and mode shapes of the structure, is found in a number of cycles with an algebraic eigenvalue problem to be solved within each cycle.

Modeling and analysis of a cracked composite cantilever beam vibrating in coupled bending and torsion

In this present study, natural frequencies of the first two modes of bending vibration for the cracked simply supported Euler–Bernoulli beam is determined using finite element analysis (FEA). FEA natural frequencies for the cracked beam are used to investigate the behavior of the cracked beam and also used in the inverse problem of crack depth detection. Dynamic behavior of the cracked simply supported beam is observed, and it is found that normalized mode shape at crack location has great effect on amount of decreasing of natural frequencies. When normalized mode shape at crack location is increased, then natural frequencies decrease. In this study, pattern of mode shape played a vital role in decreasing or increasing natural frequencies. At the midpoint of the beam, there is largest bending moment in first bending mode and there is nodal point in second bending mode. Harmonic analysis for the cracked simply supported beam is carried out to find von Mises stress responses and appearance of peaks at frequency of first bending mode is noticed in graphs of von Mises stress response, expressing high values of von Mises stress at crack tip. Inverse problem of assessing the crack depth is performed using results of FEA first mode frequency ratio and published experimental results and the method showed good results in case of high crack depth ratios.

Trivalent chromium (Cr) is an environmental contaminant, which is extensively used in tanning industries throughout the world and causes various forms of health hazards in tannery workers. Therefore, a cross-sectional study design was used to evaluate the DNA damage and oxidative stress condition in tannery workers exposed to Cr in North India. The study population comprised 100 male tanners in the exposed group and 100 healthy males (no history of Cr exposure) in the comparable control group. Baseline characteristics including age, smoking, alcohol consumption habits and duration of exposure were recorded via interviewing the subjects. Blood Cr level (measured by atomic absorption spectrophotometry), DNA damage (measured by comet assay) and oxidative stress parameters (malondialdehyde (MDA), glutathione (GSH) and superoxide dismutase (SOD)) were estimated in both the groups. As a result of statistical analysis, exposed group showed significantly higher level of Cr (p<0.0001), DNA damage (p<0.0001), MDA (p<0.0001), SOD (p<0.05) and lower level of GSH (p<0.001) when compared with controls. Smoking, alcohol consumption habits and age had no significant effect (p>0.05) on DNA damage and oxidative stress parameters in both the groups. In simple and multiple correlation analysis, DNA damage and oxidative stress parameters showed significant correlation with Cr level and duration of exposure in exposed group. The findings of the present study revealed that chronic occupational exposure to trivalent Cr may cause DNA damage and oxidative stress in tannery workers.